Question

We have three identical and independent temperature sensors that will rigger in: 90% of the cases where the temperature is high . 5% of the cases where the temperature is nominal . 1% of the cases where the temperature is low The probability of high temperature is 20%, nominal temperature is 70%, and low temperature is 10%. Describe a Bayesian network and corresponding queries for computing the following:

(a) Probability that the first sensor will trigger given that the other two sensors have also triggered

(b) Probability that the temperature is high given that all three sensors have triggered.

(c) Probability that the temperature is high given that at least one sensor has triggered.

Answer 1

let the 3 temperature sensors be A,B,C. they are identical and independent.

let X be the event that sensor is triggered.

Y1 be the event that temperature is high, Y2 be the event that temperature is nominal , Y3 be the event that temperature is low

now P[X|Y1]=0.9 P[X|Y2]=0.05 P[X|Y3]=0.01

P[Y1]=0.2 P[Y2]=0.7 P[Y3]=0.1

a) probability that the first sensor will trigger given that the other two sesnors have also triggered.

now since the sensors are independent, hence whether the first sensor will trigger or not will not depend on the other two sensors.

so probability that the first sensor will trigger given that the other two sesnors have also triggered= probability that the first sensor will trigger=P[X]=P[X and Y1]+P[X and Y2]+P[X and Y3]

=P[X|Y1]*P[Y1]+P[X|Y2]*P[Y2]+P[X|Y3]*P[Y3]=0.9*0.2+0.05*0.7+0.01*0.1=0.216 [by law of total probability]

b) P[Y1| all 3 sensors have triggered]=P[Y1 and all 3 sensors have triggered]/P[all 3 sensors have triggered] [ by theorem of conditional probability]

=P[Y1 and sensor A is triggered]*P[Y1 and sensor B is triggered]*P[Y1 and sesnor C is triggered]/P[all 3 sensors have triggered] [as the sesnsors are independent]

=P[Y1 and X]³/P[X]³ as the sensors are identical

={P[X|Y1]*P[Y1]}³/P[X]³=(0.9*0.2)³/0.216³=0.5787 [answer]